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General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Truthfulness and frugality ratio in the cheapest path auctions Estonian Theory Days 2009 Nick Gravin (based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) St.Petersburg Department of Steklov Mathematical Institute RAS, Nanyang Technological University, Singapore October 2009 Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions General set system auctions. Model: There are given a set of agents E = {1, 2, . . . , n} and a family of feasible sets F ⊆ 2E . Auctioneer is intent on hiring a team (some feasible set) of agents — winning set. Each agent i has a true cost ci , but at the auction he can bid another price bi ≥ ci . Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he loses. Every agent is selﬁsh and wants to maximize ui . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions General set system auctions. Model: There are given a set of agents E = {1, 2, . . . , n} and a family of feasible sets F ⊆ 2E . Auctioneer is intent on hiring a team (some feasible set) of agents — winning set. Each agent i has a true cost ci , but at the auction he can bid another price bi ≥ ci . Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he loses. Every agent is selﬁsh and wants to maximize ui . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions General set system auctions. Model: There are given a set of agents E = {1, 2, . . . , n} and a family of feasible sets F ⊆ 2E . Auctioneer is intent on hiring a team (some feasible set) of agents — winning set. Each agent i has a true cost ci , but at the auction he can bid another price bi ≥ ci . Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he loses. Every agent is selﬁsh and wants to maximize ui . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions General set system auctions. Model: There are given a set of agents E = {1, 2, . . . , n} and a family of feasible sets F ⊆ 2E . Auctioneer is intent on hiring a team (some feasible set) of agents — winning set. Each agent i has a true cost ci , but at the auction he can bid another price bi ≥ ci . Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he loses. Every agent is selﬁsh and wants to maximize ui . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions General set system auctions. Model: There are given a set of agents E = {1, 2, . . . , n} and a family of feasible sets F ⊆ 2E . Auctioneer is intent on hiring a team (some feasible set) of agents — winning set. Each agent i has a true cost ci , but at the auction he can bid another price bi ≥ ci . Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he loses. Every agent is selﬁsh and wants to maximize ui . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions First price rule Auction is called a ﬁrst price if auctioneer always chooses the cheapest feasible set, i.e. a set F with minimal e∈F b(e). Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions examples Non-trivial issues concerning computational complexity appears when feasible sets are given implicitly. 1 Path auction: agents — edges in the graph; feasible sets — all paths between two given vertices s and t. 2 k-Path: agents — edges in the graph; feasible sets — k edge-disjoint paths from s to t. 3 Vertex cover auction: agents — vertices; feasible sets — all vertex covers of edges. 4 Spanning tree auction: agents — edges; feasible sets — spanning trees. 5 Perfect matching auction: agents — edges in bipartite graph; feasible sets — perfect matchings. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions examples Non-trivial issues concerning computational complexity appears when feasible sets are given implicitly. 1 Path auction: agents — edges in the graph; feasible sets — all paths between two given vertices s and t. 2 k-Path: agents — edges in the graph; feasible sets — k edge-disjoint paths from s to t. 3 Vertex cover auction: agents — vertices; feasible sets — all vertex covers of edges. 4 Spanning tree auction: agents — edges; feasible sets — spanning trees. 5 Perfect matching auction: agents — edges in bipartite graph; feasible sets — perfect matchings. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions examples Non-trivial issues concerning computational complexity appears when feasible sets are given implicitly. 1 Path auction: agents — edges in the graph; feasible sets — all paths between two given vertices s and t. 2 k-Path: agents — edges in the graph; feasible sets — k edge-disjoint paths from s to t. 3 Vertex cover auction: agents — vertices; feasible sets — all vertex covers of edges. 4 Spanning tree auction: agents — edges; feasible sets — spanning trees. 5 Perfect matching auction: agents — edges in bipartite graph; feasible sets — perfect matchings. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions examples Non-trivial issues concerning computational complexity appears when feasible sets are given implicitly. 1 Path auction: agents — edges in the graph; feasible sets — all paths between two given vertices s and t. 2 k-Path: agents — edges in the graph; feasible sets — k edge-disjoint paths from s to t. 3 Vertex cover auction: agents — vertices; feasible sets — all vertex covers of edges. 4 Spanning tree auction: agents — edges; feasible sets — spanning trees. 5 Perfect matching auction: agents — edges in bipartite graph; feasible sets — perfect matchings. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions examples Non-trivial issues concerning computational complexity appears when feasible sets are given implicitly. 1 Path auction: agents — edges in the graph; feasible sets — all paths between two given vertices s and t. 2 k-Path: agents — edges in the graph; feasible sets — k edge-disjoint paths from s to t. 3 Vertex cover auction: agents — vertices; feasible sets — all vertex covers of edges. 4 Spanning tree auction: agents — edges; feasible sets — spanning trees. 5 Perfect matching auction: agents — edges in bipartite graph; feasible sets — perfect matchings. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions examples Non-trivial issues concerning computational complexity appears when feasible sets are given implicitly. 1 Path auction: agents — edges in the graph; feasible sets — all paths between two given vertices s and t. 2 k-Path: agents — edges in the graph; feasible sets — k edge-disjoint paths from s to t. 3 Vertex cover auction: agents — vertices; feasible sets — all vertex covers of edges. 4 Spanning tree auction: agents — edges; feasible sets — spanning trees. 5 Perfect matching auction: agents — edges in bipartite graph; feasible sets — perfect matchings. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Nash equilibrium in ﬁrst price auction Deﬁnition Nash equilibrium(NE) in the ﬁrst price auction is an assignment of agent bids b, such that no agent e can increase its utility ue by varying its bid be . NE exists if there is no agent belonging to all feasible sets(monopolist). We can ﬁnd a NE explicitly in polynomial time. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Nash equilibrium in ﬁrst price auction Deﬁnition Nash equilibrium(NE) in the ﬁrst price auction is an assignment of agent bids b, such that no agent e can increase its utility ue by varying its bid be . NE exists if there is no agent belonging to all feasible sets(monopolist). We can ﬁnd a NE explicitly in polynomial time. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Nash equilibrium in ﬁrst price auction Deﬁnition Nash equilibrium(NE) in the ﬁrst price auction is an assignment of agent bids b, such that no agent e can increase its utility ue by varying its bid be . NE exists if there is no agent belonging to all feasible sets(monopolist). We can ﬁnd a NE explicitly in polynomial time. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Truthful mechanisms Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Mechanisms for winners and payments selection Suppose the true costs of all agents are private. Agents have only one chance to send their bids to the auctioneer. Auctioneer (buyer, center) on the base of these bids should choose the winning set and how much to pay to them. Deﬁnition The mechanism of winners selection and their payments is called truthful if each agent maximizes utility by bidding its true cost (for any ﬁxed bids of other agents). Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Mechanisms for winners and payments selection Suppose the true costs of all agents are private. Agents have only one chance to send their bids to the auctioneer. Auctioneer (buyer, center) on the base of these bids should choose the winning set and how much to pay to them. Deﬁnition The mechanism of winners selection and their payments is called truthful if each agent maximizes utility by bidding its true cost (for any ﬁxed bids of other agents). Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Mechanisms for winners and payments selection Suppose the true costs of all agents are private. Agents have only one chance to send their bids to the auctioneer. Auctioneer (buyer, center) on the base of these bids should choose the winning set and how much to pay to them. Deﬁnition The mechanism of winners selection and their payments is called truthful if each agent maximizes utility by bidding its true cost (for any ﬁxed bids of other agents). Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Mechanisms for winners and payments selection Suppose the true costs of all agents are private. Agents have only one chance to send their bids to the auctioneer. Auctioneer (buyer, center) on the base of these bids should choose the winning set and how much to pay to them. Deﬁnition The mechanism of winners selection and their payments is called truthful if each agent maximizes utility by bidding its true cost (for any ﬁxed bids of other agents). Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Example (Vickrey) Second price auction (for particular set system). There are n sellers and auctioneer needs to bay exactly one thing. Mechanism selects the cheapest agent and pays to him the bid of the second by the price agent. Is it clear why this mechanism is truthful? Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Example (Vickrey) Second price auction (for particular set system). There are n sellers and auctioneer needs to bay exactly one thing. Mechanism selects the cheapest agent and pays to him the bid of the second by the price agent. Is it clear why this mechanism is truthful? Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Example (Vickrey-Clarke-Groves) VCG for general set system. Choose the cheapest feasible set F and pay independently to each agent in F its threshold bid, i.e. such bid that F is still the cheapest feasible set. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions VCG for cheapest path c: 0 0 0 0 0 S 1 T Initial true costs. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions VCG for cheapest path c: 0 0 0 0 0 S 1 T Cheapest path. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions VCG for cheapest path b: 1 5 1 1 5 5 1 1 5 5 S 1 T Cheapest Nash equilibrium. Total payment is 1. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions VCG for cheapest path c: 0 1 0 0 0 S 1 T Payment to each agent on the winning path should be 1. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions VCG for cheapest path c: 1 1 1 1 1 S 1 T The total payment is 5, but for NE it was only 1. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions How much mechanism overpays (E, F) — monopoly free set system; M — truthful mechanism for (E, F). c is a true cost vector; ν(c) — the total payment in the cheapest Nash for (E, F, c); PM (c) — total payment of M for c; Deﬁnition Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a mechanism eﬃciency. PM (c) ΦM = sup . c ν(c) Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions How much mechanism overpays (E, F) — monopoly free set system; M — truthful mechanism for (E, F). c is a true cost vector; ν(c) — the total payment in the cheapest Nash for (E, F, c); PM (c) — total payment of M for c; Deﬁnition Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a mechanism eﬃciency. PM (c) ΦM = sup . c ν(c) Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions How much mechanism overpays (E, F) — monopoly free set system; M — truthful mechanism for (E, F). c is a true cost vector; ν(c) — the total payment in the cheapest Nash for (E, F, c); PM (c) — total payment of M for c; Deﬁnition Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a mechanism eﬃciency. PM (c) ΦM = sup . c ν(c) Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions How much mechanism overpays (E, F) — monopoly free set system; M — truthful mechanism for (E, F). c is a true cost vector; ν(c) — the total payment in the cheapest Nash for (E, F, c); PM (c) — total payment of M for c; Deﬁnition Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a mechanism eﬃciency. PM (c) ΦM = sup . c ν(c) Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions How much mechanism overpays (E, F) — monopoly free set system; M — truthful mechanism for (E, F). c is a true cost vector; ν(c) — the total payment in the cheapest Nash for (E, F, c); PM (c) — total payment of M for c; Deﬁnition Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a mechanism eﬃciency. PM (c) ΦM = sup . c ν(c) Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Motivation for frugality ratio ΦM ≥ 1, since one can take c to be by itself a Nash equilibrium and then PM (c) ≥ ν(c). ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 ) (E, F) is a Matroid iﬀ for every two sets S, T ∈ F, there is a bijection f between S \ T and T \ S such that S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Motivation for frugality ratio ΦM ≥ 1, since one can take c to be by itself a Nash equilibrium and then PM (c) ≥ ν(c). ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 ) (E, F) is a Matroid iﬀ for every two sets S, T ∈ F, there is a bijection f between S \ T and T \ S such that S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Motivation for frugality ratio ΦM ≥ 1, since one can take c to be by itself a Nash equilibrium and then PM (c) ≥ ν(c). ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 ) (E, F) is a Matroid iﬀ for every two sets S, T ∈ F, there is a bijection f between S \ T and T \ S such that S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Mechanism design for cheapest path auctions Known results: VCG — ΦVCG for example with long path and an edge is quadratic in terms of the optimal frugality ratio. √ KKT -mechanism gives a linear approximation to the optimal frugality ratio (Φ√ ≤ 2X and any truthful 1 mechanism has frugality ratio at least √2 X ). Pruning-lifting mechanism (our paper) for k-path auctions provides optimal frugality ratio. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Mechanism design for cheapest path auctions Known results: VCG — ΦVCG for example with long path and an edge is quadratic in terms of the optimal frugality ratio. √ KKT -mechanism gives a linear approximation to the optimal frugality ratio (Φ√ ≤ 2X and any truthful 1 mechanism has frugality ratio at least √2 X ). Pruning-lifting mechanism (our paper) for k-path auctions provides optimal frugality ratio. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Mechanism design for cheapest path auctions Known results: VCG — ΦVCG for example with long path and an edge is quadratic in terms of the optimal frugality ratio. √ KKT -mechanism gives a linear approximation to the optimal frugality ratio (Φ√ ≤ 2X and any truthful 1 mechanism has frugality ratio at least √2 X ). Pruning-lifting mechanism (our paper) for k-path auctions provides optimal frugality ratio. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions KKT mechanism for cheapest path √ -mechanism for cheapest path auctions. We will see that for some graphs it will have much better frugality ratio than VCG. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions First step. Find two edge-disjoint paths P, P minimizing b(P) + b(P ). (Ignore the rest of the graph) S T Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Second step. Let s = v1 , v2 , . . . , vk+1 = t be the vertices that P, P have in common, in the order in which they appear in P and P . Let Pi (resp. Pi ) be the subpath of P (resp. P ) from vi to vi+1 . v1 v2 v2 v3 Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Third step. For each i, include Pi in the solution iﬀ |Pi |b(Pi ) ≤ |Pi |b(Pi ); otherwise, include Pi . b: 0 0 3 v1 5 v2 v2 v3 1 1 0 4 1 Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Forth step. Pay to each winner its threshold bid (i.e. the largest value that he can bid and still win, if the others bid the same). b: 4 2 −3 3 3 3 − √ 4 2 v1 3 3 v2 2 3 4 v2 v3 1 1 0 4 1 Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Bounds on Φ√ √ -mechanism is truthful, thus b = c. Φ√ ≤ 2 maxi |Pi ||Pi |. √ Thus for the edge and a path graph we have Φ√ = O(2 l), while ΦVCG = O(l). 1 There is a lower bound √ 2 maxi |Pi ||Pi | on any truthful mechanism, thus 1 √ max |Pi ||Pi | ≤ Φ√ ≤ 2 max |Pi ||Pi |. 2 i i Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Bounds on Φ√ √ -mechanism is truthful, thus b = c. Φ√ ≤ 2 maxi |Pi ||Pi |. √ Thus for the edge and a path graph we have Φ√ = O(2 l), while ΦVCG = O(l). 1 There is a lower bound √ 2 maxi |Pi ||Pi | on any truthful mechanism, thus 1 √ max |Pi ||Pi | ≤ Φ√ ≤ 2 max |Pi ||Pi |. 2 i i Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Bounds on Φ√ √ -mechanism is truthful, thus b = c. Φ√ ≤ 2 maxi |Pi ||Pi |. √ Thus for the edge and a path graph we have Φ√ = O(2 l), while ΦVCG = O(l). 1 There is a lower bound √ 2 maxi |Pi ||Pi | on any truthful mechanism, thus 1 √ max |Pi ||Pi | ≤ Φ√ ≤ 2 max |Pi ||Pi |. 2 i i Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions Bounds on Φ√ √ -mechanism is truthful, thus b = c. Φ√ ≤ 2 maxi |Pi ||Pi |. √ Thus for the edge and a path graph we have Φ√ = O(2 l), while ΦVCG = O(l). 1 There is a lower bound √ 2 maxi |Pi ||Pi | on any truthful mechanism, thus 1 √ max |Pi ||Pi | ≤ Φ√ ≤ 2 max |Pi ||Pi |. 2 i i Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions VCG Nash equilibriums frugality ratio Truthful mechanisms √ KKT -mechanism k-paths and cheapest path auctions k-paths and cheapest path auctions k-paths and cheapest path auctions Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Nash for k-paths Theorem Any Nash equilibrium for the cheapest path auction with respect to the bid vector b should have two non-intersecting by the edges cheapest paths. Theorem Any Nash equilibrium for the k-paths auction with respect to the bid vector b should have k + 1 non-intersecting by the edges cheapest paths. These theorems are non trivial results of graph theory. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Nash for k-paths Theorem Any Nash equilibrium for the cheapest path auction with respect to the bid vector b should have two non-intersecting by the edges cheapest paths. Theorem Any Nash equilibrium for the k-paths auction with respect to the bid vector b should have k + 1 non-intersecting by the edges cheapest paths. These theorems are non trivial results of graph theory. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Nash for k-paths Theorem Any Nash equilibrium for the cheapest path auction with respect to the bid vector b should have two non-intersecting by the edges cheapest paths. Theorem Any Nash equilibrium for the k-paths auction with respect to the bid vector b should have k + 1 non-intersecting by the edges cheapest paths. These theorems are non trivial results of graph theory. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Our mechanism for k-path auctions Pruning-lifting mechanism Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Pruning step Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that δ(P1 , . . . , Pk+1 ) = max b(P) P∈∪Pi is minimized. (Ignore the rest of the graph.) It is NP hard to ﬁnd P1 , . . . , Pk+1 even with < k + 1 factor approximation of δ(P1 , . . . , Pk+1 ) to the optimal one. The cheapest k + 1 ﬂow gives k + 1 approximation. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Pruning step Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that δ(P1 , . . . , Pk+1 ) = max b(P) P∈∪Pi is minimized. (Ignore the rest of the graph.) It is NP hard to ﬁnd P1 , . . . , Pk+1 even with < k + 1 factor approximation of δ(P1 , . . . , Pk+1 ) to the optimal one. The cheapest k + 1 ﬂow gives k + 1 approximation. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Pruning step Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that δ(P1 , . . . , Pk+1 ) = max b(P) P∈∪Pi is minimized. (Ignore the rest of the graph.) It is NP hard to ﬁnd P1 , . . . , Pk+1 even with < k + 1 factor approximation of δ(P1 , . . . , Pk+1 ) to the optimal one. The cheapest k + 1 ﬂow gives k + 1 approximation. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Lifting step We construct a graph H as follows: we take arcs of ∪i Pi as vertices for H; we draw an edge between e and e iﬀ there is no path from s to t containing both e, e . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Lifting step We construct a graph H as follows: we take arcs of ∪i Pi as vertices for H; we draw an edge between e and e iﬀ there is no path from s to t containing both e, e . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Lifting step We construct a graph H as follows: we take arcs of ∪i Pi as vertices for H; we draw an edge between e and e iﬀ there is no path from s to t containing both e, e . Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Lifting step ∪Pi s t Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Lifting step ∪Pi 11 00 00 11 00 11 00 11 11 00 11 00 00 11 11 00 11 00 11 00 00 11 s 11 00 000 111 111 000 111 000 111 000 00 11 00 11 00 11 t 00 11 11 00 11 00 11 00 11 00 00 11 11 00 11 00 00 11 11 00 00 11 00 11 11 00 Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Lifting step H 1111111111111 000000000000011 00 000000 0000000000000000000000 111111 1111111111111111111111 00000000 11111111 11 00 0000000000000 1111111111111 0000000000000000000000 000000 111111 1111111111111111111111 00000000 11111111 0000000000000 111111111111111 00 00 11 0 1 00000000000 11111111111 000000 0000000000000000000000 111111 1111111111111111111111 00000000 11111111 0000000000000 1111111111111 11 00 0 1 00000000000 11111111111 000000 0000000000000000000000 1111111111111111111111 111111 00000000 11111111 1111111111111 0000000000000 00000000 11 00 0 1 11111111111 00000000000 000000 0000000000000000000000 1111111111111111111111 111111 11111111 0000000000000 1111111111111 0 1 00000000000 11111111111 0000000000000000000000 000000 1111111111111111111111 111111 00000000 11111111 1111111111111 0000000000000 000000 0000000000000000000000 111111 00000000 11111111 11 00 000 111 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1111111111111 111 00000 11 0 1 00000000000 11111111111 00 11 0000000000000000000000 000000 1111111111111111111111 111111 00000000 11111111 1111111111111 0000000000000 111 00000 11 0 1 11111111111 00000000000 11 00 000000 0000000000000000000000 111111 1111111111111111111111 00000000 11111111 111 00000 11 0 1 00000000000 11111111111 s 00000 11111 11 00 11111 00000 11111 00000 00000 11111 00000 11111 11111 00000 0000000000000000000000 000000 1111111111111111111111 111111 00000000 11111111 000000 111111 000 111 0000000000000000000000 1111111111111111111111 00000000 11111111 000000 0000000000000000000000 1111111111111111111111 111111 00000000 11111111 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000 111 111 00 000 11 000 111 0000000000000000000000 1111111111111111111111 000 111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 00 11 00 11 000 11 111 00 0000 1111 111 11 000 00 0000 1111 111 00 000 11 1111 0000 11 00 11111111111 00000000000 00 11 00 11 0 1 0 1 00000000000 11111111111 0 1 00000000000 11111111111 0 1 00000000000 11111111111 0 1 00000000000 11111111111 t 11111 00000 0000000000000000000000 000000 111111 1111111111111111111111 00000000 11111111 1111 0000 00 11 0 1 11111111111 00000000000 00000 11111 0000000000000000000000 000000 1111111111111111111111 111111 00000000 11111111 1111 0000 00 11 0 1 11111111111 00000000000 0 1 00000 11111 0000000000000000000000 000000 111111 1111111111111111111111 00000000 11111111 1111 0000 00 11 00000000000 11111111111 0 1 00000000000 11111111111 11111 00000 000000 0000000000000000000000 111111 1111111111111111111111 00000000 11111111 0000 1111 00 11 0 1 00000000000 11111111111 11111 00000 000000 0000000000000000000000 1111111111111111111111 111111 00000000 11111111 1111 0000 00 11 0 1 00000000000 11111111111 00000 11111 00000 11111 000000000000000000 111111111111111111 0000000000000000000000 000000 111111 1111111111111111111111 00000000 11111111 00 11 0000 1111 00 11 0 1 00000000000 11111111111 00000 1111100000000 11111111 000000000000000000 111111111111111111 0000000000000000000000 000000 111111 1111111111111111111111 00 11 0000 1111 00 11 111111111111111111 000000000000000000 0 1 00000000000 11111111111 00000 11111 0000000000000000000000 1111111111111111111111 00000000 11111111 11 00 0000 1111 00 11 111111111111111111 000000000000000000 0 1 00000000000 11111111111 11111 00000 0000000000000000000000 1111111111111111111111 00000000 11111111 1111 0000 00 11 000000000000000000 11111111111111111100 11 0 1 00000000000 11111111111 00000 11111 0000000000000000000000 1111111111111111111111 00000000 11111111 00000000 1111 0000 00 11 11 00 0 1 00000000000 11111111111 00000 1111111111111 00000000 11 00 0 1 00000000000 11111111111 00000 1111111111111 00000000 11111111 0 1 00000000000 11111111111 00000 1111100000000 11111111 0 1 11111111111 00000000000 0 00 11 00000 1111100000000 11111111 1 00000000000 11111111111 0 1 00 11 00000 1111100000000 11111111 00000000000 11111111111 00 11 0 1 11 00 00000000000 11111111111 11 00 00 11 00 11 Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Weighting step We split H into its components of connectivity H1 , . . . , Hl . Let Ai be an adjacency matrix of Hi . For each i we ﬁnd the positive eigenvector (wi ) and eigenvalue αi of the matrix Ai . Let α = maxi αi . (Ai ) (wi ) = αi (wi ) b(e) Deﬁne b (e) = w (e) . Let P the maximum weight path with respec to b (). Take ∪Pi \ P as the set of winners. Pay to each agent its threshold bid. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Weighting step We split H into its components of connectivity H1 , . . . , Hl . Let Ai be an adjacency matrix of Hi . For each i we ﬁnd the positive eigenvector (wi ) and eigenvalue αi of the matrix Ai . Let α = maxi αi . (Ai ) (wi ) = αi (wi ) b(e) Deﬁne b (e) = w (e) . Let P the maximum weight path with respec to b (). Take ∪Pi \ P as the set of winners. Pay to each agent its threshold bid. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Weighting step We split H into its components of connectivity H1 , . . . , Hl . Let Ai be an adjacency matrix of Hi . For each i we ﬁnd the positive eigenvector (wi ) and eigenvalue αi of the matrix Ai . Let α = maxi αi . (Ai ) (wi ) = αi (wi ) b(e) Deﬁne b (e) = w (e) . Let P the maximum weight path with respec to b (). Take ∪Pi \ P as the set of winners. Pay to each agent its threshold bid. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions Weighting step We split H into its components of connectivity H1 , . . . , Hl . Let Ai be an adjacency matrix of Hi . For each i we ﬁnd the positive eigenvector (wi ) and eigenvalue αi of the matrix Ai . Let α = maxi αi . (Ai ) (wi ) = αi (wi ) b(e) Deﬁne b (e) = w (e) . Let P the maximum weight path with respec to b (). Take ∪Pi \ P as the set of winners. Pay to each agent its threshold bid. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions ΦPL α , ΦPL = k where α is a lower bound on the frugality ratio of any truthful k mechanism. So PL is the optimal mechanism with respect to frugality ratio. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov) General auctions Nash equilibriums Truthful mechanisms k-paths and cheapest path auctions ΦPL α , ΦPL = k where α is a lower bound on the frugality ratio of any truthful k mechanism. So PL is the optimal mechanism with respect to frugality ratio. Truthfulness and frugality ratio in the cheapest path auctions Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)

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posted: | 11/19/2012 |

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